If it did happen, it would cost a fortune to get up there. You're talking a trade on par with the Atlanta-Cleveland trade (27 to 6) that landed Julio Jones. That trade cost Atlanta 5 draft picks. Atlanta gave up a 1st, 2nd, and 4th in 2011 and a 1st and 4th in 2012 in that trade. Without a 2nd rounder, KC could be looking at about a 1st and 3rd in 2014 and a 1st and 3rd in 2015 and that would be to get to 6. Even if Clowney is the type of player that can make a huge impact, you still lose out what could be 3 additional starters when you talk about giving up those kinds of picks. I wouldn't do it, it is too expensive. If I'm a team sitting in the top 15 with all of my picks in both drafts, then I'd probably make the move since it would likely only cost me a 1st, 2nd, and 1st in 2015 on the higher end.
In any event, I suspect Clowney is going to either Houston or St. Louis and I'm thinking Houston isn't going to pass on that type of talent no matter how bad they might need a QB.